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Kiril DATCHEV & András VASY
Semiclassical resolvent estimates at trapped sets
Tome 62, no 6 (2012), p. 2379-2384.
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Ann. Inst. Fourier, Grenoble
62, 6 (2012) 2379-2384
SEMICLASSICAL RESOLVENT ESTIMATES
AT TRAPPED SETS
by Kiril DATCHEV & András VASY (*)
Abstract. — We extend our recent results on propagation of semiclassical
resolvent estimates through trapped sets when a priori polynomial resolvent bounds
hold. Previously we obtained non-trapping estimates in trapping situations when
the resolvent was sandwiched between cutoffs χ microlocally supported away from
the trapping: ‖χRh(E + i0)χ‖ = O(h−1), a microlocal version of a result of Burq
and Cardoso-Vodev. We now allow one of the two cutoffs, χ̃, to be supported at
the trapped set, giving ‖χRh(E+ i0)χ̃‖ = O(
√
a(h)h−1) when the a priori bound
is ‖χ̃Rh(E + i0)χ̃‖ = O(a(h)h−1).
Résumé. — Nous étendons nos résultats récents sur la propagation d’esti-
mations de résolvantes semi-classiques à travers des ensembles captifs sous des
bornes a priori de type polynomial. Précédemment, nous obtenions des estima-
tions non-captives dans des situations captives quand la résolvante est contrôlée
par au dessus et en dessous par des fonctions cutoff χ dont le support micro-
local est situé loin de l’ensemble captif : ‖χRh(E + i0)χ‖ = O(h−1) (version
microlocale d’un résultat de Burq et Cardoso-Vodev). Nous considérons mainte-
nant le cas où l’une des deux fonctions cutoff, χ̃, est à support dans l’ensemble
captif, obtenant ‖χRh(E + i0)χ̃‖ = O(
√
a(h)h−1) lorsque la borne a priori est
‖χ̃Rh(E + i0)χ̃‖ = O(a(h)h−1).
This short article is an addendum to the previous paper by K. Datchev
and A. Vasy.
Let (X, g) be a Riemannian manifold which is asymptotically conic or
asymptotically hyperbolic in the sense of [6], let V ∈ C∞0 (X) be real valued,
let P = h2∆g + V (x), where ∆g > 0, and fix E > 0.
Keywords: Resolvent estimates, trapping, propagation of singularities.
Math. classification: 58J47, 35L05.
(*) The first author is partially supported by a National Science Foundation postdoc-
toral fellowship, and the second author is partially supported by the National Science
Foundation under grants DMS-0801226 and DMS-1068742.
2380 Kiril DATCHEV & András VASY
Theorem 1. — [6, Theorem 1.2] Suppose that for any χ0 ∈ C∞0 (X)
there exist C0, k, h0 > 0 such that for any ε > 0, h ∈ (0, h0] we have
‖χ0(h2∆g + V − E − iε)−1χ0‖L2(X)→L2(X) 6 C0h−k. (1)
Let KE ⊂ T ∗X be the set of trapped bicharacteristics at energy E, and
suppose that b ∈ C∞0 (T ∗X) is identically 1 near KE . Then there exist
C1, h1 > 0 such that for any ε > 0, h ∈ (0, h1] we have the following
nontrapping estimate:
‖〈r〉−1/2−δ(1−Op(b))(h2∆g + V − E − iε)−1(1−Op(b))〈r〉−1/2−δ‖L2(X)→L2(X)
6 C1h
−1. (2)
Here by bicharacteristics at energy E we mean integral curves in p−1(E)
of the Hamiltonian vector field Hp of the Hamiltonian p = |ξ|2 +V (x), and
the trapped ones are those which remain in a compact set for all time. We
use the notation r = r(z) = dg(z, z0), where dg is the distance function on
X induced by g and z0 ∈ X is fixed but arbitrary.
If KE = ∅ then (1) holds with k = 1. If KE 6= ∅ but the trapping is
sufficiently ‘mild’, then (1) holds for some k > 1: see [6] for details and
examples. The point is that the losses in (1) due to trapping are removed
when the resolvent is cutoff away from KE . Theorem 1 is a more precise
and microlocal version of an earlier result of Burq [1] and Cardoso and
Vodev [3], but the assumption (1) is not needed in [1, 3]. See [6] for addi-
tional background and references for semiclassical resolvent estimates and
trapping.
In this paper we prove that an improvement over the a priori estimate
(1) holds even when one of the factors of (1−Op(b)) is removed:
Theorem 2. — Suppose that there exist k > 0 and a(h) 6 h−k such
that for any χ0 ∈ C∞0 (X) there exists h0 > 0 such that for any ε > 0,
h ∈ (0, h0] we have
‖χ0(h2∆g + V − E − iε)−1χ0‖L2(X)→L2(X) 6 a(h)/h. (3)
Suppose that b ∈ C∞0 (T ∗X) is identically 1 near KE . Then there exist
C1, h1 > 0 such that for any ε > 0, h ∈ (0, h1],
‖〈r〉−1/2−δ(1−Op(b))(h2∆g + V − E − iε)−1〈r〉−1/2−δ‖L2(X)→L2(X)
6 C1
√
a(h)/h. (4)
Note that by taking adjoints, analogous estimates follow if 1 − Op(b) is
placed to the other side of (h2∆g + V − E − iε)−1.
ANNALES DE L’INSTITUT FOURIER
SEMICLASSICAL RESOLVENT ESTIMATES AT TRAPPED SETS 2381
Such results were proved by Burq and Zworski [2, Theorem A] and Chris-
tianson [4, (1.6)] when KE consists of a single hyperbolic orbit. Theorem 2
implies an optimal semiclassical resolvent estimate for the example opera-
tor of [6, §5.3]: it improves [6, (5.5)] to
‖χ0(P − λ)−1χ0‖ 6 C log(1/h)/h.
Further, this improved estimate can be used to extend polynomial resolvent
estimates from complex absorbing potentials to analogous estimates for
damped wave equations; this is a result of Christianson, Schenk, Wunsch
and the second author [5].
Theorems 1 and 2 follow from microlocal propagation estimates in a
neighborhood of KE , or more generally in a neighborhood of a suitable
compact invariant subset of a bicharacteristic flow.
To state the general results, suppose X is a manifold, P ∈ Ψm,0(X) a self
adjoint, order m > 0, semiclassical pseudodifferential operator on X, with
principal symbol p. For I ⊂ R compact and fixed, denote the characteristic
set by Σ = p−1(I), and suppose that the projection to the base, π : Σ→ X,
is proper (it is sufficient, for example, to have p classically elliptic). Suppose
that Γ b T ∗X is invariant under the bicharacteristic flow in Σ. Define the
forward, resp. backward flowout Γ+, resp. Γ−, of Γ as the set of points
ρ ∈ Σ, from which the backward, resp. forward bicharacteristic segments
tend to Γ, i.e. for any neighborhood O of Γ there exists T > 0 such that
−t > T , resp. t > T , implies γ(t) ∈ O, where γ is the bicharacteristic with
γ(0) = ρ. Here we think of Γ as the trapped set or as part of the trapped
set, hence points in Γ−, resp. Γ+ are backward, resp. forward, trapped.
Suppose V , W are neighborhoods of Γ with V ⊂W , W compact. Suppose
also that
If ρ ∈W \ Γ+, resp. ρ ∈W \ Γ−,
then the backward, resp. forward bicharacteristic from ρ intersects W \V .
(5)
This means that all bicharacterstics in V which stay in V for all time tend
to Γ.
The main result of [6], from which the other results in the paper follow,
is the following:
Theorem 3. — [6, Theorem 1.3] Suppose that ‖u‖H−N
h
6 h−N for some
N ∈ N and (P − λ)u = f , Reλ ∈ I and Imλ > −O(h∞). Suppose f is
O(1) on W , WFh(f) ∩ V = ∅, and u is O(h−1) on W ∩ Γ− \ V . Then u is
O(h−1) on W ∩ Γ+ \ Γ.
TOME 62 (2012), FASCICULE 6
2382 Kiril DATCHEV & András VASY
Here we say that u is O(a(h)) at ρ ∈ T ∗X if there exists B ∈ Ψ0,0(X)
elliptic at ρ with ‖Bu‖L2 = O(a(h)). We say u is O(a(h)) on a set E ⊂ T ∗X
if it is O(a(h))) at each ρ ∈ E.
Note that there is no conclusion on u at Γ; typically it will be merely
O(h−N ) there. However, to obtain O(h−1) bounds for u on Γ+ we only
needed to assume O(h−1) bounds for u on Γ− and nowhere else. Note also
that by the propagation of singularities, if u is O(h−1) at one point on any
bicharacteristic, then it is such on the whole forward bicharacteristic. If
| Imλ| = O(h∞) then the same is true for backward bicharacteristics.
In this paper we show that a (lesser) improvement on the a priori bound
holds even when f is not assumed to vanish microlocally near Γ:
Theorem 4. — Suppose that ‖u‖H−N
h
6 h−N for some N ∈ N and
(P −λ)u = f , Reλ ∈ I and Imλ > −O(h∞). Suppose f is O(1) on W , u is
O(a(h)h−1) onW , and u is O(h−1) onW ∩Γ−\V . Then u is O(
√
a(h)h−1)
on W ∩ Γ+ \ Γ.
In [6] Theorem 1 is deduced from Theorem 3. Theorem 2 follows from
Theorem 4 by the same argument.
Proof of Theorem 4. — The argument is a simple modification of the
argument of [6, End of Section 4, Proof of Theorem 1.3]; we follow the
notation of this proof. Recall first from [6, Lemma 4.1] that if U− is a
neighborhood of (Γ− \ Γ) ∩ (W \ V ) then there is a neighborhood U ⊂ V
of Γ such that if α ∈ U \ Γ+ then the backward bicharacteristic from α
enters U−. Thus, if one assumes that u is O(h−1) on Γ− and f is O(1) on
V , it follows that that u is O(h−1) on U \Γ+, provided U− is chosen small
enough that u is O(h−1) on U−. Note also that, because U ⊂ V , f is O(1)
on U . We will show that u is O(
√
a(h)h−1) on U ∩ Γ+ \ Γ: the conclusion
on the larger set W ∩ Γ+ \ Γ follows by propagation of singularities.
Next, [6, Lemma 4.3] states that if U1 and U0 are open sets with Γ ⊂
U1 b U0 b U then there exists a nonnegative function q ∈ C∞0 (U) such
that
q = 1 near Γ, Hpq 6 0 near Γ+, Hpq < 0 on ΓU0+ \ U1.
Moreover, we can take q such that both √q and
√
−Hpq are smooth
near Γ+.
Remark. — The last paragraph in the proof of [6, Lemma 4.3] should
be replaced by the following: To make
√
−Hpq̃ smooth, let ψ(s) = 0 for
s 6 0, ψ(s) = e−1/s for s > 0, and assume as we may that Uρ ∩ Sρ is
a ball with respect to a Euclidean metric (in local coordinates near ρ) of
ANNALES DE L’INSTITUT FOURIER
SEMICLASSICAL RESOLVENT ESTIMATES AT TRAPPED SETS 2383
radius rρ > 0 around ρ. We then choose ϕρ to behave like ψ(r′ρ
2 − |.|2)
with r′ρ < rρ for |.| close to r′ρ, bounded away from 0 for smaller values of
|.|, and choose −χ′ρ to vanish like ψ at the boundary of its support. That
sums of products of such functions have smooth square roots follows from
[7, Lemma 24.4.8].
The proof of Theorem 4 proceeds by induction: we show that if u is O(hk)
on a sufficiently large compact subset of U ∩ Γ+ \ Γ, then u is O(hk+1/2)
on ΓU0+ \ U1, provided
√
a(h)h−1 6 Chk+1/2.
Now let U− be an open neighborhood of Γ+∩ supp q which is sufficiently
small that Hpq 6 0 on U− and that
√
−Hpq is smooth on U−. Let U+ be
an open neighborhood of supp q \U− whose closure is disjoint from Γ+ and
from T ∗X \ U . Define φ± ∈ C∞(U+ ∪ U−) with suppφ± ⊂ U± and with
φ2+ + φ2− = 1 near supp q.
Put
b
def= φ−
√
−Hpq2, e
def= φ2+Hpq2.
Let Q,B,E ∈ Ψ−∞,0(X) have principal symbols q, b, e, and microsupports
supp q, supp b, supp e, so that
i
h
[P,Q∗Q] = −B∗B + E + hF,
with F ∈ Ψ−∞,0(X) such that WF′h F ⊂ supp dq ⊂ U \ Γ. But
i
h
〈[P,Q∗Q]u, u〉 = 2
h
Im〈Q∗Q(P − λ)u, u〉+ 2
h
〈Q∗Q Imλu, u〉
> −2h−1‖Q(P − λ)u‖ ‖Qu‖ − O(h∞)‖u‖2 > −Ch−2a(h)−O(h∞),
where we used Imλ > −O(h∞) and that on supp q, (P − λ)u is O(1). So
‖Bu‖2 6 〈Eu, u〉+ h〈Fu, u〉+ Ch−2a(h) +O(h∞).
But |〈Eu, u〉| 6 Ch−2 because WF′hE ∩ Γ+ = ∅ gives that u is O(h−1) on
WF′hE by the first paragraph of the proof. Meanwhile |〈Fu, u〉| 6 C(h−2 +
h2k) because all points of WF′h F are either in U\Γ+, where we know u is
O(h−1) from the first paragraph of the proof, or on a single compact subset
of U∩Γ+\Γ, where we know that u is O(hk) by inductive hypothesis. Since
b =
√
−Hpq2 > 0 on ΓU0+ \ U1, we can use microlocal elliptic regularity to
conclude that u is O(hk+1/2) on ΓU0+ \ U1, as desired. 
TOME 62 (2012), FASCICULE 6
2384 Kiril DATCHEV & András VASY
BIBLIOGRAPHY
[1] N. Burq, “Lower bounds for shape resonances widths of long range Schrödinger
operators”, Amer. J. Math. 124 (2002), no. 4, p. 677-755.
[2] N. Burq & M. Zworski, “Geometric control in the presence of a black box”, J. Amer.
Math. Soc. 17:2 (2004), no. 4, p. 443-471.
[3] F. Cardoso & G. Vodev, “Uniform estimates of the resolvent of the Laplace-
Beltrami operator on infinite volume Riemannian manifolds. II”, Ann. Henri Poincaré
3 (2002), no. 4, p. 673-691.
[4] H. Christianson, “Semiclassical non-concentration near hyperbolic orbits”, J.
Funct. Anal. 246 (2007), no. 2, p. 145-195, Corrigendum, J. Funct. Anal., 258 (2010),
no. 3 p. 1060-1065.
[5] H. Christianson, E. Schenck, A. Vasy & J. Wunsch, “From resolvent estimates to
damped waves”, To appear in J. Anal. Math. Preprint available at arXiv:1206.1565,
2012.
[6] K. Datchev & A. Vasy, “Propagation through trapped sets and semiclassical resol-
vent estimates”, Annales de l’Institut Fourier 62.6 (2012), p. 2345-2375.
[7] L. Hörmander, The Analysis of Linear Partial Differential Operators. III. Pseudo-
Differential Operators, Springer Verlag, 1994.
Manuscrit reçu le 12 juin 2012,
accepté le 10 janvier 2013.
Kiril DATCHEV
Department of Mathematics,
Massachusetts Institute of Technology,
Cambridge, MA 02139-4397,
U.S.A.
datchev@math.mit.edu
András VASY
Department of Mathematics,
Stanford University,
Stanford, CA 94305-2125,
U.S.A.
andras@math.stanford.edu
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